Memory effects in atmospheric methane chemistry: A Riemann–Liouville fractional framework with Monte Carlo validation

We develop a fractional calculus framework for atmospheric methane chemistry that incorporates memory effects arising from the complex, multiscale nature of tropospheric transport and mixing. Using the Riemann–Liouville fractional derivative, we derive analytical expressions for methane concentration response to emission perturbations, showing that the classical feedback factor f = ( 1 − R ) − 1 emerges as the long-time limit of a time-dependent effective feedback governed by the Mittag-Leffler function. The framework is rigorously validated through Monte Carlo simulation of the underlying continuous-time random walk process using the subordinator approach, achieving mean relative errors below 2% across fractional orders α ∈ [ 0 . 5 , 0 . 9 ] and over three decades in time. Application to chemistry–climate model results suggests weak but measurable non-Markovian behavior with α ≈ 0 . 85 –0.95, indicating that perturbation recovery may be 30%–150% slower than classical exponential predictions. For stronger memory effects ( α ≈ 0 . 75 ), equilibration times can exceed Markovian estimates by a factor of five or more. This framework provides a physically-grounded parameterization for non-exponential relaxation in atmospheric chemistry models, with implications for understanding the recent acceleration of methane-climate feedbacks and optimizing mitigation strategies. • Fractional derivative framework for methane chemistry incorporates memory effects. • Mittag-Leffler function governs time-dependent feedback approaching classical limit. • Monte Carlo subordination method validates theory with <2% mean error. • Atmospheric methane shows weak non-Markovian behavior ( α ≈ 0.85–0.95). • Memory effects extend equilibration times 30%–150% beyond exponential estimates.